N. S. Witte, C. M. Ormerod
We construct a Lax pair for the $ E^{(1)}_6 $ $q$-Painlev\'e system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices \cite{Wi_2010a}. Our study treats one special case of such lattices - the $q$-linear lattice - through a natural generalisation of the big $q$-Jacobi weight. As a by-product of our construction we derive the coupled first order $q$-difference equations for the $ E^{(1)}_6 $ $q$-Painlev\'e system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations.
View original:
http://arxiv.org/abs/1207.0041
No comments:
Post a Comment